The present invention relates generally to nonlinear systems and more particularly to methods and systems for processing signals generated from nonlinear systems.
Virtually all practically engineered and/or natural systems experience nonlinear behavior. As defined herein, a nonlinear system is a system which exhibits a nonlinear relationship between its input and output (i.e., the system fails to obey the principal of superposition between its input and output). Examples of systems which exhibit nonlinear behavior include, inter alia, most biological systems, fluid flow systems, optical systems, imaging systems, RF receiver and transmitter systems, magnetic devices and magnetic recording systems, analog electronic systems, amplifier systems, loud speaker systems, radar systems and sonar systems.
A signal output from a nonlinear system typically includes a nonlinear component. As defined herein, use of the term signal in conjunction with nonlinear systems is meant to denote both (1) the mathematical description of any measurable phenomena in nature or in human-made systems and (2) the mathematically described function of one or more variable depending on one or more parameters. Examples of signals include, inter alia, light intensity, voltage, pressure, magnetic field strength and electric field strength.
Nonlinearities inherent in a particular system often introduce substantial nonlinear distortion products (e.g., harmonics, intermodulations, spurs, etc.) into the output signal generated by said system. In turn, the introduction of these nonlinear distortion products may limit the ability of signal processors to separate the desired component of the output signal (typically the linear component) therefrom, thereby effectively compromising the overall performance of the system, which is highly undesirable.
As an example, in the field of digital signal communications, an analog-to-digital converter (a device commonly used to convert a signal from the analog domain to the digital domain) as well as traditional front-end devices (e.g., signal amplifiers, analog filters, etc.) often introduce nonlinear irregularities into the communication signal. As will be described further below, the introduction of nonlinear irregularities (also referred to herein as nonlinear distortions) into an output signal effectively limits the overall performance of said nonlinear communication devices.
As can be appreciated, there presently exist a number of well-established devices and mathematical modeling tools for analyzing and treating the linear irregularities of an output signal. However, contrary to its linear counterparts, there presently exist limited devices and mathematical modeling tools for analyzing and treating the nonlinear irregularities of an output signal. In fact, most known mathematical means for processing nonlinear signal irregularities tend to be specialized for particular types of nonlinear systems. In addition, even when said mathematical modeling tools exist, they are often very cumbersome in nature and computationally intensive, which is highly undesirable.
Referring now to FIG. 1, there is shown a graphical representation of a digital output signal that has been generated by a nonlinear system (e.g., a receiver that includes an analog-to-digital converter), the output signal being represented in terms of power as a function of frequency. It should be noted that the graphical representation shown in FIG. 1 is not an actual measured output signal but rather a simplified version of a sample output signal that is useful in understanding the principals of the present invention. It should also be noted that FIG. 1 represents a sample output signal produced in response to the application of a two tone input signal into a nonlinear system. However, it is to be understood that alternative types of input signals (i.e., other than of the two tone variety) would produce similar types of output signals when introduced into the same nonlinear system.
As can be seen, the output signal shown in FIG. 1 includes a representation of the original two tone input signal, the representation of the original two tone input signal being identified by reference numeral 11. However, it should be noted that the nonlinear system additionally introduces noise 13 and nonlinear intermodulation products 15 into the output signal. As defined herein, the term “intermodulation products” relates to the nonlinear distortions that are produced in response to the application of a signal into a nonlinear system.
It should be noted that intermodulation products 15 decrease the portion of the two tone input signal 11 which can be utilized for processing, the strength of the usable portion of the input signal 11 being commonly used to define the dynamic range for the device. Accordingly, it is to be understood that it is a primary object in digital signal processing to treat an output signal with nonlinear irregularities in such as manner so as to reduce the strength of intermodulation products 15 without reducing the strength of the two tone input 11, thereby maximizing the intermodulation-free dynamic range (IFDR). As can be appreciated, the ability to increase the usable portion of the two-tone input signal (i.e., IFDR) has a number of practical applications in the fields of, but not limited to, radar systems, sonar systems and digital communication systems (e.g., increasing signal strength in cellular applications).
In recent years, polynomial difference equation (PDE) filters, and in particular Volterra filters, have been used for nonlinear system representations, in large part because the output is a linear function of the filters parameters. While Volterra filters have been advantageous for nonlinear system representations, there is a disadvantage inherent to this filter representation. Specifically, the disadvantage inherent to the Volterra representation, and other similar nonlinear system representations, is the computational complexity involved in calculating the output. As the polynomial order or memory increases, the number of parameters in the Volterra filter increases rapidly and thus, the number of computations rapidly becomes prohibitive.
In U.S. Pat. No. 6,639,537 to G. M. Raz, the disclosure of which is incorporated herein by reference, there is disclosed a highly linear analog-to-digital (ADC) conversion system that has an analog front-end device in cascade with a standard ADC converter, and a tunable digital nonlinear equalizer. The equalizer corrects the quantization distortion, deviations from ideal response, and additive noises generated by the analog front-end device and ADC converter. The equalizer is formed by three main parts: a generate function streams unit (GFSU), finite inpulse response (FIR) filters and a summer. The equalizer receives the unequalized output from the ADC converter and generates a plurality of monomial streams in a systolic fashion. Each of the monomial streams is passed through a corresponding linear finite impulse response FIR filter. A sum of all outputs from the FIR filters produces a unique equalized output with the nonlinear distortion reduced to a satisfactory level. The FIR filter coefficients are determined by an identity equalizer coefficient unit (IECU), and test signal generator with different types of test signals. The FIR filter coefficients are set to minimize an error function.
Although useful in reducing nonlinear distortions, systems of the type as described above in the '537 patent suffer from a couple notable shortcomings.
As a first drawback, the equalization coefficient for each of the linear filters in the nonlinear equalizer described in the '537 patent has linear relationship with the output signal but, in contrast thereto, has a nonlinear relationship with the input signal. Because the equalization coefficients for the linear filters have a nonlinear correspondence with the input signal, it is to be understood that linear mathematical tools can not be used to perform certain essential calculations, which is highly undesirable.
As a second drawback, the nonlinear equalizer described in the '537 patent provides no means for partitioning an input signal into a plurality of subspace components, thereby limiting its performance in certain applications. For instance, the nonlinear equalizer described in the '537 patent can not perform linear equalization of a wide-band system using narrow-band filtering tools, which is highly undesirable.